On Triangular Sum Labeling of Graphs

GRD Journals- Global Research and Development Journal for Engineering | Volume 3 | Issue 7 | June 2018 ISSN: 2455-5703

On Triangular Sum Labelings of Graphs Shankaran P Department of Mathematics N.M.A.M. Institute of Technology, Nitte-574 110, Karnataka, India

Abstract Let G = (V,E) be a (p, q)-graph. A graph G is said to admit a triangular sum labeling, if its vertices can be labeled by non-negative integers so that the values on the edges, obtained as the sum of the labels of their end vertices, are the first q triangular numbers.

In this paper, we obtain a necessary condition for an eulerian graph to admit a triangular sum labeling and show that some classes of graphs admit a triangular sum labeling. Also we show that some classes of graphs can be embedded as an induced subgraph of a triangular sum graph.

Keywords- Triangular Numbers, Triangular Sum Labeling/Graphs, Dutch Windmill, Locally Finite Tree 2000 Mathematics Subject Classification: 05C78

I. INTRODUCTION Several practical problems in real-life situations have motivated the study of labelings of graphs which are required to obey a variety of conditions depending on the structure of graphs. There is an enormous amount of literature built upon several kinds of labelings over the past four decades or so and for a survey of results on labelings we refer to [2].

For various graph theoretical notations and terminology we refer to Harary [3] and West [10].

Definition 1.1. [4]. When n copies of K 3 share a common vertex, then the resulting graph is called a dutch windmill, denoted as DW(n).

II. MAIN RESULTS Definition 2.1. Let G = (V, E) be a graph with p vertices and q edges. Let Ti be the ith triangular number given by Ti = i(i+1)/2 (see [1] ). A triangular sum labeling of a graph G is a one-to-one function f: V(G) N (where N is the set of non-negative integers) that induces a bijection f+: E(G)  T1,T2,…,Tq defined by f+(uv)= f(u)+f(v),  e = uv E(G) . The graph which admits such a labeling is called a triangular sum graph.

We adopt the following notation throughout this paper. f(G) =  f(u)/u  V(G) and f+(G) =  f +(e)/e  E(G).

Example 2.2. The path Pn = (v1, v2,…, vn) is a triangular sum graph.

The function f: V(Pn)  N defined by f(v1) = 0 and f(vi) = Ti-1- f(vi-1), 2  i  n is such that f+(Pn) = T1, T2,…, Tn-1. Triangular sum labeling of P6 is given in Figure 1.

Fig. 1: Triangular sum labeling of P6

Example 2.3. The star K1,n is a triangular sum graph. If V(K1,n ) = {v,v1, v2,…, vn} with deg( v) = n, then f defined by f(v) = 0 and f(vi) = Ti is a triangular sum labeling of K1,n.

Theorem 2.4. If G is an eulerian (p,q)-graph admitting a triangular sum labeling, then q ≡ 0,2,3,4,5,6,7,8,9,10 or 11 (mod 12).

Proof. Let G be an eulerian (p,q)-graph admitting a triangular sum labeling f. Since an eulerian graph can be decomposed into edge-disjoint cycles, the sum of the edge values is always even. Further the sum of the first q triangular numbers is

Hence q(q+1)(q+2) is a multiple of 12 and hence the result follows Theorem 2.5. Any tree obtained from a star K1,n by extending each edge to a path is a triangular sum graph.

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On Triangular Sum Labelings of Graphs (GRDJE/ Volume 3 / Issue 7 / 006)

Proof. Let K1,n be the given star with the root vertex u1,0 and T be a rooted tree obtained by replacing each edge by a path (u10, n

ui1, ui2,…, ui mi). Note that the tree T contains

m i 1

i

edges and n paths each of length at least two.

We consider three cases.

Case 1. All the mi’s are distinct.

Let m1> m2 >…> mn  2 and rj be the number of vertices at jth level, 1  j  m1.

Clearly 1 rm1  rm1 1 … r2  r1= n. Define a labeling f:V(T)  N by f(u1,0) = 0 f(ui,1) = Ti , 1  i  n and j 1

f(ui,j) = T

ik

–f(ui,j-1), 1 i  rj, 2  j  m1, k =

r t 1

t

Then one can easily check that the vertex labels are all in increasing order and f+(T) = {T1,T2,…, Tq}. Hence the tree T is a triangular sum graph.

Case 2. All the mi’s are equal.

Let mi = m, 1  i  n. Define a mapping f:V(T)N by f(u1,0) = 0, f(ui,1) = Ti , 1  i  n and f(ui,j) = Ti+(j-1)n –f(ui,j-1), 1 i  n, 2  j  m.

Then one can see that the vertex labels are all in increasing order and f+(T ) is the set of triangular numbers T1,T2,…,Tq .

Case 3. m1 m2 … mn  2 and not all mi’s are equal.

Let 1 rm1  rm1 1 … r2 = r1= n, where rj be the number of vertices at jth level. Define, f(u1,m1) = 0, f(u , ) = T j –f(u ), 1  j  m 1 m1-,j

1, m1-,j+1

1

f(ui , 1) = Tm1 i 1 –f(u1,0), 2  i  n j 1

f(ui ,j) = Tm1  k i  j –f(ui ,j-1), 2≤ i ≤ rj, 2 j  m2 and k =

r t 1

t

Then f(ui+1,j) > f(ui,j),  i,j and f+(T) = {T1,T2,…, Tq} Example 2.6. The tree given in Figure 2a is obtained by replacing each edge of the star K1,3 by paths of different length and the tree given in Figure 2b is obtained by replacing each edge of the star K1,4 by paths of equal length. The triangular sum labelings of the trees are also given.

Fig. 2 (a)

Fig. 2 (b)

Theorem 2.7. The lobster T obtained by joining the centres of k copies of a star to a new vertex w is a triangular sum graph.

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On Triangular Sum Labelings of Graphs (GRDJE/ Volume 3 / Issue 7 / 006)

Proof. Let T be the lobster obtained by joining the centre of k stars K I,n. Denote the root vertex of ith star K 1, n as wi, i=1,2,…,k, the pendant vertices of ith star as vi,j , i=1,2,…,k, j=1,2,…,n. Note that T contains nk+k edges. Define a labeling f:V(T) N by f(w) =1, f(wi) = Ti –1 , 1  i  k and f(v ) = Tk  j  m -f(w ) , 1  i  k,1  j  n, m = (i-1)n. i,j

i

Since f(vi,j) > f(vi,j+1) for some i, j, we have f(vi,j) + f(wi) > f(vi,j+1) + f(wi) and one can see that f+(T) = T1,T2,…,Tq. Thus T is a triangular sum graph.

Fig. 3: An example of a triangular sum labeling of a lobster

Theorem 2.8. The complete n-ary tree Tm of level m is a triangular sum graph.

Proof. Denote the root vertex of Tm as u0 ; the vertices of level 1 as u11,u12,…, u1n1 ; that of level 2 as u21,u22,…, u 2n 2 ;…; the m

vertices of level m as um1,um2,…, u mn m . Note that Tm contains

n i 1

i

edges. Define the map f:V(Tm)  N by

f(u0 ) = 0, f(u1, j ) = Tj , 1  j  n1 i 1

f(ui, j ) = Tt+ j -f(ui-1,r ), 2  i m, 1  j  n , where t = i

n

k

k 1

r =1 if 1  j  n r = 2 if n+1  j  2n r = 3 if 2n+1  j  3n … … … … r =nm-1 if nm-n+1  j  nm Since f(ui+1,j+1) > f(ui+1,j), it follows that f(ui,r) + f(ui+1,j) are all distinct and f+(Tm)=T1,T2,…,Tq.

Example 2.9. An infinite graph G is said to be locally finite if every vertex of G has finite degree. We now give an example of a locally finite tree which admits a triangular sum labeling. Consider the locally finite tree given in Figure 4. Let {ui: i ≥1} be the vertices of the infinite path in T and let vi be a pendant vertex adjacent to ui. Then f defined by f(u1) = T1, f(u2) = 0, f(ui) = T2i-2-f(ui-1), i ≥3, f(v1) = T2-1 and f(vi) = T2i-1-f(ui), i ≥2 is a triangular sum labeling of G.

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On Triangular Sum Labelings of Graphs (GRDJE/ Volume 3 / Issue 7 / 006)

Fig. 4: Locally finite tree T

We propose the following conjecture.

Conjecture 2.10. All trees are triangular sum graphs.

Theorem 2.11. The complete graph Kn is triangular sum if and only if n  2.

Proof. Trivially K1 and K2 are triangular sum graphs.

Assume n  3. Assign the numbers 0 and 1 to any two vertices of Kn. To get the next triangular number 3, we have to assign the numbers either 2 or 3 to a third vertex. In both cases, we get a non-triangular number 2 or 4 as an edge label, and hence Kn is not a triangular sum graph if n  3.

Theorem 2.12. The dutch windmill DW (n) is not a triangular sum graph.

Proof. Denote the root vertex of the dutch windmill DW(n) as u and the remaining two vertices of ith triangle as v2i-1 and v2i for 1 ≤ i ≤ n. Suppose DW(n) admits a triangular sum labeling f.

Case 1. f(u) = 0.

Then all vi’s, i=1,2,…,2n must be assigned triangular numbers. However, 1+Ti  Tj, for all i and j and hence at least one of the induced edge values is a non-triangular number, which is a contradiction.

Case 2. f(v1) = 0.

Then the two adjacent vertices of v1 must be assigned the triangular numbers 1 and Ti , i >1, again leading to a contradiction.

III. EMBEDDINGS OF TRIANGULAR SUM GRAPHS In this section we show that the complete graphs K3 and K4 can be embedded as induced sub graphs of triangular sum graphs. We also prove that the dutch windmill DW (n) can be embedded as an induced sub graph of a triangular sum graph.

Theorem 3.1. Let G be a unicyclic graph consisting of a unique triangle (v1,v2,v3,v1) with deg(v2) = deg(v3) = 2, a path P = v1,u1,u2,…,un of length n and k pendant vertices w1,w2,…,wk adjacent to v1. Then G is a triangular sum graph for all n  3.

Proof. Define f:V(G)  N by f(v1) = 0, f(vi) = T2i-1, i = 2,3. f(u1) = 1 f(ui) = T2(i-1) -f(ui-1), i = 2,3. f(ui) = Ti+3-f(ui-1), 4 ≤ i ≤ n and f(wi) = Tn+3+i , 1≤ i≤ k.

Clearly f+(G) ={T1,T2 ,…, Tq}and hence G is a triangular sum graph.

Lemma 3.2. The complete graph K4 can be embedded as an induced subgraph of a triangular sum graph.

Proof. Let V(K4)={v1, v2, v3, v4}. Let G be the graph obtained by attaching 40 pendant vertices vi, 4≤ i ≤ 43 to v1. Then f defined by f(v1) = 0, f(v2) = T9 = 45, f(v3) = T13 = 91, f(v4) = T16 = 990 and f(v5,v6,…,v43) = {T1,T2,…,T43}-{T9,T13,T16} is a triangular sum labeling of G. It is checked, with the help of a computer, that there are 446 triangular numbers within the range [1,105]. Also we found that there does not exist 4 triangular numbers, which on pairwise addition gives triangular numbers. Hence we strongly believe that the complete graph K5 is a forbidden subgraph for a triangular sum graph.

Conjecture 3.3. The complete graph Kn, n ≥ 5 is a forbidden subgraph for a triangular sum graph.

Theorem 3.4. The dutch windmill DW(n) can be embedded as an induced subgraph of a triangular sum graph.

Proof. Denote the root vertex of the dutch windmill DW(n) as u and the remaining two vertices of ith triangle as v2i-1 and v2i for 1 ≤ i ≤ n respectively. Consider the equation i2+j2+i+j-k-k2 = 24 (1) Let (xi, yi, zi), where xi< yi< zi,1 ≤ i ≤ n be the first n solutions of equation (1) [viz. (6, 8, 9),(7, 15, 16), (10, 20, 22) etc.].

Define, f(u) = 6,

 6 ,1  in X f(v2i-1) = T I

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On Triangular Sum Labelings of Graphs (GRDJE/ Volume 3 / Issue 7 / 006)

 6 , 1  i n . y f(v2i) = T i Then one can see that the vertices u and v1 receive triangular numbers and all the remaining vertices receive non-triangular numbers by the above map f. Join two pendant vertices of K1, b, b = zn-3n, to the vertices u and v1. Without loss of generality, assign the number 0 to the root vertex and the missing triangular numbers from T1 to

Tz n

to the pendant vertices of K1,b in a

Tz n

one-to-one manner. Clearly the induced edge values of the resulting graph is {T1,T2,…, }. Hence the proof.

Given below is an example for embedding of DW (2) as an induced subgraph of a triangular sum graph.

Fig. 5: Embedding of DW (2) as an induced subgraph of a triangular su

IV. CONCLUSIONS As every graph cannot be embedded as an induced subgraph of a triangular sum graph, it is interesting to embed classes of graphs as an induced subgraph of a triangular sum graph. Also, we would like to study more on cycles admitting triangular sum labeling.

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